What is the solution for y^{''}+12y^'+ay=0 ?

The solution for y^{''}+12y^'+ay=0 is y=c_{1}e^{(-6+sqrt(-a+36))t}+c_{2}e^{(-sqrt(-a+36)-6)t}

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y^{''}+12y^'+ay=0

y^{^{''}} + 12 y^{^{'}} + a y = 0

y = c_{1} e^{(- 6 + - a + 36) t} + c_{2} e^{(- - a + 36 - 6) t}

Solution steps

y^{''} (t) + 12 y^{'} (t) + a y (t) = 0

Solve linear ODE:

y = c_{1} e^{(- 6 + - a + 36) t} + c_{2} e^{(- - a + 36 - 6) t}

y = c_{1} e^{(- 6 + - a + 36) t} + c_{2} e^{(- - a + 36 - 6) t}

y^{^{''}} + 3 y^{^{'}} + 2 y = 0, y (0) = 1, y^{^{'}} (0) = 0

\frac{d ^{2} y}{d x ^{2}} + 5^{2} y = 0

y^{^{''}} + 6 y^{^{'}} + 13 y = 0, y (0) = 3

2 y^{^{''}} + 9 y^{^{'}} - 5 y = 0

y^{^{''}} + 1 y = 0, y (π^{2}) = 3, y^{^{'}} (π^{2}) = - 1

What is the solution for y^{''}+12y^'+ay=0 ?
The solution for y^{''}+12y^'+ay=0 is y=c_{1}e^{(-6+sqrt(-a+36))t}+c_{2}e^{(-sqrt(-a+36)-6)t}